Monopole Floer homology and invariant theta characteristics

Abstract

We describe a relationship between the monopole Floer homology of three-manifolds and the geometry of Riemann surfaces. For an automorphism $\phi$ of a compact Riemann surface $\Sigma$ with quotient $P^1$, there is a natural correspondence between theta characteristics $L$ on $\Sigma$ which are invariant under $\phi$ and self-conjugate ${\text{spin}}^c$ structures $s_L$ on the mapping torus $M_{\phi }$ of $\phi$. We show that the monopole Floer homology groups of $(M_{\phi },s_L)$ are explicitly determined by the eigenvalues of the (lift of the) action of $\phi$ on $H^0\left(L\right)$, the space of holomorphic sections of $L$, and discuss several consequences of this description. Our result is based on a detailed analysis of the transversality properties of the Seiberg–Witten equations for suitable small perturbations.